Classifying Camina groups: a theorem of Dark and Scoppola. (Q404499)

A non-abelian finite group \(G\) is called a Camina group if the conjugacy class of every element \(g\in G\setminus G'\) is \(gG'\). Camina groups have been classified by \textit{R. Dark} and \textit{C. M. Scoppola} [J. Algebra 181, No. 3, 787-802 (1996; Zbl 0860.20017)]. However, as the author points out, a part of their argument relies on Lemma 2.1 of \textit{I. D. Macdonald}, [Isr. J. Math. 56, 335-344 (1986; Zbl 0618.20015)], whose proof has a gap (although there seems to exist a way of amending the Dark-Scoppola proof and avoiding use of that lemma).NEWLINENEWLINE In the present paper the author gives another proof of most of the classification of Camina groups, avoiding any gaps, although using in the proofs some of the facts proved in earlier papers. The new proof is based on Theorem 2, which strengthens a result of \textit{I. M. Isaacs} [Can. J. Math. 41, No. 1, 68-82 (1989; Zbl 0686.20002)] on Camina pairs.